%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET931+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:26:41 AM UTC 2025
% Result : Theorem 0.22s 0.40s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.14 % Problem : SET931+1 : TPTP v9.0.0. Released v3.2.0.
% 0.04/0.14 % Command : run_E %s %d THM
% 0.15/0.36 % Computer : n003.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Fri Jun 20 05:51:09 EDT 2025
% 0.15/0.36 % CPUTime :
% 0.22/0.40 % SZS status Theorem
% 0.22/0.40 % SZS output start Proof
% 0.22/0.40 tff(unordered_pair_type, type, (
% 0.22/0.40 unordered_pair: ( $i * $i ) > $i)).
% 0.22/0.40 tff(tptp_fun_C_2_type, type, (
% 0.22/0.40 tptp_fun_C_2: $i)).
% 0.22/0.40 tff(tptp_fun_B_3_type, type, (
% 0.22/0.40 tptp_fun_B_3: $i)).
% 0.22/0.40 tff(tptp_fun_A_4_type, type, (
% 0.22/0.40 tptp_fun_A_4: $i)).
% 0.22/0.40 tff(singleton_type, type, (
% 0.22/0.40 singleton: $i > $i)).
% 0.22/0.40 tff(empty_set_type, type, (
% 0.22/0.40 empty_set: $i)).
% 0.22/0.40 tff(set_difference_type, type, (
% 0.22/0.40 set_difference: ( $i * $i ) > $i)).
% 0.22/0.40 tff(subset_type, type, (
% 0.22/0.40 subset: ( $i * $i ) > $o)).
% 0.22/0.40 tff(1,assumption,(~(set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)), introduced(assumption)).
% 0.22/0.40 tff(2,plain,
% 0.22/0.40 ((~(((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set))) <=> (~(((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(3,plain,
% 0.22/0.40 (((~((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2)))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)) <=> (~(((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(4,plain,
% 0.22/0.40 (((~(A!4 = empty_set)) & (~(A!4 = singleton(B!3))) & (~(A!4 = singleton(C!2))) & (~(A!4 = unordered_pair(B!3, C!2)))) <=> (~((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(5,plain,
% 0.22/0.40 ((((~(A!4 = empty_set)) & (~(A!4 = singleton(B!3))) & (~(A!4 = singleton(C!2))) & (~(A!4 = unordered_pair(B!3, C!2)))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)) <=> ((~((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2)))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set))),
% 0.22/0.40 inference(monotonicity,[status(thm)],[4])).
% 0.22/0.40 tff(6,plain,
% 0.22/0.40 ((((~(A!4 = empty_set)) & (~(A!4 = singleton(B!3))) & (~(A!4 = singleton(C!2))) & (~(A!4 = unordered_pair(B!3, C!2)))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)) <=> (~(((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)))),
% 0.22/0.40 inference(transitivity,[status(thm)],[5, 3])).
% 0.22/0.40 tff(7,plain,
% 0.22/0.40 ((((~(A!4 = empty_set)) & (~(A!4 = singleton(B!3))) & (~(A!4 = singleton(C!2))) & (~(A!4 = unordered_pair(B!3, C!2)))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)) <=> (~(((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)))),
% 0.22/0.40 inference(transitivity,[status(thm)],[6, 2])).
% 0.22/0.40 tff(8,plain,
% 0.22/0.40 ((~![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> (set_difference(A, unordered_pair(B, C)) = empty_set)))) <=> (~![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> (set_difference(A, unordered_pair(B, C)) = empty_set))))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(9,plain,
% 0.22/0.40 ((~![A: $i, B: $i, C: $i] : ((set_difference(A, unordered_pair(B, C)) = empty_set) <=> (~((((~(A = empty_set)) & (~(A = singleton(B)))) & (~(A = singleton(C)))) & (~(A = unordered_pair(B, C))))))) <=> (~![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> (set_difference(A, unordered_pair(B, C)) = empty_set))))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(10,axiom,(~![A: $i, B: $i, C: $i] : ((set_difference(A, unordered_pair(B, C)) = empty_set) <=> (~((((~(A = empty_set)) & (~(A = singleton(B)))) & (~(A = singleton(C)))) & (~(A = unordered_pair(B, C))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t75_zfmisc_1')).
% 0.22/0.40 tff(11,plain,
% 0.22/0.40 (~![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> (set_difference(A, unordered_pair(B, C)) = empty_set)))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.22/0.40 tff(12,plain,
% 0.22/0.40 (~![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> (set_difference(A, unordered_pair(B, C)) = empty_set)))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[11, 8])).
% 0.22/0.40 tff(13,plain,
% 0.22/0.40 (~![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> (set_difference(A, unordered_pair(B, C)) = empty_set)))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.22/0.40 tff(14,plain,
% 0.22/0.40 (~![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> (set_difference(A, unordered_pair(B, C)) = empty_set)))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[13, 8])).
% 0.22/0.40 tff(15,plain,(
% 0.22/0.40 ((~(A!4 = empty_set)) & (~(A!4 = singleton(B!3))) & (~(A!4 = singleton(C!2))) & (~(A!4 = unordered_pair(B!3, C!2)))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)),
% 0.22/0.40 inference(skolemize,[status(sab)],[14])).
% 0.22/0.40 tff(16,plain,
% 0.22/0.40 (~(((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[15, 7])).
% 0.22/0.40 tff(17,plain,
% 0.22/0.40 ((((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)) | ((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) | (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)),
% 0.22/0.40 inference(tautology,[status(thm)],[])).
% 0.22/0.40 tff(18,plain,
% 0.22/0.40 (((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) | (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)),
% 0.22/0.40 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.22/0.40 tff(19,plain,
% 0.22/0.40 ((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))),
% 0.22/0.40 inference(unit_resolution,[status(thm)],[18, 1])).
% 0.22/0.40 tff(20,plain,
% 0.22/0.40 (^[A: $i, B: $i] : refl(((set_difference(A, B) = empty_set) <=> subset(A, B)) <=> ((set_difference(A, B) = empty_set) <=> subset(A, B)))),
% 0.22/0.40 inference(bind,[status(th)],[])).
% 0.22/0.40 tff(21,plain,
% 0.22/0.40 (![A: $i, B: $i] : ((set_difference(A, B) = empty_set) <=> subset(A, B)) <=> ![A: $i, B: $i] : ((set_difference(A, B) = empty_set) <=> subset(A, B))),
% 0.22/0.40 inference(quant_intro,[status(thm)],[20])).
% 0.22/0.40 tff(22,plain,
% 0.22/0.40 (![A: $i, B: $i] : ((set_difference(A, B) = empty_set) <=> subset(A, B)) <=> ![A: $i, B: $i] : ((set_difference(A, B) = empty_set) <=> subset(A, B))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(23,axiom,(![A: $i, B: $i] : ((set_difference(A, B) = empty_set) <=> subset(A, B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t37_xboole_1')).
% 0.22/0.40 tff(24,plain,
% 0.22/0.40 (![A: $i, B: $i] : ((set_difference(A, B) = empty_set) <=> subset(A, B))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.22/0.40 tff(25,plain,(
% 0.22/0.40 ![A: $i, B: $i] : ((set_difference(A, B) = empty_set) <=> subset(A, B))),
% 0.22/0.40 inference(skolemize,[status(sab)],[24])).
% 0.22/0.40 tff(26,plain,
% 0.22/0.40 (![A: $i, B: $i] : ((set_difference(A, B) = empty_set) <=> subset(A, B))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.22/0.40 tff(27,plain,
% 0.22/0.40 ((~![A: $i, B: $i] : ((set_difference(A, B) = empty_set) <=> subset(A, B))) | ((set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set) <=> subset(A!4, unordered_pair(B!3, C!2)))),
% 0.22/0.40 inference(quant_inst,[status(thm)],[])).
% 0.22/0.40 tff(28,plain,
% 0.22/0.40 ((set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set) <=> subset(A!4, unordered_pair(B!3, C!2))),
% 0.22/0.40 inference(unit_resolution,[status(thm)],[27, 26])).
% 0.22/0.41 tff(29,plain,
% 0.22/0.41 ((~((set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set) <=> subset(A!4, unordered_pair(B!3, C!2)))) | (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set) | (~subset(A!4, unordered_pair(B!3, C!2)))),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(30,plain,
% 0.22/0.41 ((set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set) | (~subset(A!4, unordered_pair(B!3, C!2)))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[29, 28])).
% 0.22/0.41 tff(31,plain,
% 0.22/0.41 (~subset(A!4, unordered_pair(B!3, C!2))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[30, 1])).
% 0.22/0.41 tff(32,plain,
% 0.22/0.41 (^[A: $i, B: $i, C: $i] : refl((((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C))) <=> (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(33,plain,
% 0.22/0.41 (![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C))) <=> ![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[32])).
% 0.22/0.41 tff(34,plain,
% 0.22/0.41 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(trans(monotonicity(rewrite(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> (~((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)))), ((((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))) <=> ((~((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set))) <=> subset(A, unordered_pair(B, C))))), rewrite(((~((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set))) <=> subset(A, unordered_pair(B, C))) <=> (~(((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C))))), ((((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))) <=> (~(((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))))), rewrite((~(((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))) <=> (~(((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C))))), ((((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))) <=> (~(((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))))), ((~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C)))) <=> (~(~(((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C))))))), rewrite((~(~(((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C))))) <=> (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))), ((~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C)))) <=> (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(35,plain,
% 0.22/0.41 (![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C)))) <=> ![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[34])).
% 0.22/0.41 tff(36,plain,
% 0.22/0.41 (![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C)))) <=> ![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(37,plain,
% 0.22/0.41 (^[A: $i, B: $i, C: $i] : trans(trans(monotonicity(rewrite((~((((~(A = empty_set)) & (~(A = singleton(B)))) & (~(A = singleton(C)))) & (~(A = unordered_pair(B, C))))) <=> (~((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))))), ((subset(A, unordered_pair(B, C)) <=> (~((((~(A = empty_set)) & (~(A = singleton(B)))) & (~(A = singleton(C)))) & (~(A = unordered_pair(B, C)))))) <=> (subset(A, unordered_pair(B, C)) <=> (~((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))))))), rewrite((subset(A, unordered_pair(B, C)) <=> (~((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))))) <=> (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))))), ((subset(A, unordered_pair(B, C)) <=> (~((((~(A = empty_set)) & (~(A = singleton(B)))) & (~(A = singleton(C)))) & (~(A = unordered_pair(B, C)))))) <=> (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C)))))), rewrite((~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C)))) <=> (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))))), ((subset(A, unordered_pair(B, C)) <=> (~((((~(A = empty_set)) & (~(A = singleton(B)))) & (~(A = singleton(C)))) & (~(A = unordered_pair(B, C)))))) <=> (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(38,plain,
% 0.22/0.41 (![A: $i, B: $i, C: $i] : (subset(A, unordered_pair(B, C)) <=> (~((((~(A = empty_set)) & (~(A = singleton(B)))) & (~(A = singleton(C)))) & (~(A = unordered_pair(B, C)))))) <=> ![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[37])).
% 0.22/0.41 tff(39,axiom,(![A: $i, B: $i, C: $i] : (subset(A, unordered_pair(B, C)) <=> (~((((~(A = empty_set)) & (~(A = singleton(B)))) & (~(A = singleton(C)))) & (~(A = unordered_pair(B, C))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','l46_zfmisc_1')).
% 0.22/0.41 tff(40,plain,
% 0.22/0.41 (![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.22/0.41 tff(41,plain,
% 0.22/0.41 (![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[40, 36])).
% 0.22/0.41 tff(42,plain,(
% 0.22/0.41 ![A: $i, B: $i, C: $i] : (~(((~(A = empty_set)) & (~(A = singleton(B))) & (~(A = singleton(C))) & (~(A = unordered_pair(B, C)))) <=> subset(A, unordered_pair(B, C))))),
% 0.22/0.41 inference(skolemize,[status(sab)],[41])).
% 0.22/0.41 tff(43,plain,
% 0.22/0.41 (![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[42, 35])).
% 0.22/0.41 tff(44,plain,
% 0.22/0.41 (![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[43, 33])).
% 0.22/0.41 tff(45,plain,
% 0.22/0.41 (((~![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))) | (((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> subset(A!4, unordered_pair(B!3, C!2)))) <=> ((~![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))) | (((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> subset(A!4, unordered_pair(B!3, C!2))))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(46,plain,
% 0.22/0.41 ((((A!4 = unordered_pair(B!3, C!2)) | (A!4 = singleton(C!2)) | (A!4 = singleton(B!3)) | (A!4 = empty_set)) <=> subset(A!4, unordered_pair(B!3, C!2))) <=> (((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> subset(A!4, unordered_pair(B!3, C!2)))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(47,plain,
% 0.22/0.41 (((~![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))) | (((A!4 = unordered_pair(B!3, C!2)) | (A!4 = singleton(C!2)) | (A!4 = singleton(B!3)) | (A!4 = empty_set)) <=> subset(A!4, unordered_pair(B!3, C!2)))) <=> ((~![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))) | (((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> subset(A!4, unordered_pair(B!3, C!2))))),
% 0.22/0.41 inference(monotonicity,[status(thm)],[46])).
% 0.22/0.41 tff(48,plain,
% 0.22/0.41 (((~![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))) | (((A!4 = unordered_pair(B!3, C!2)) | (A!4 = singleton(C!2)) | (A!4 = singleton(B!3)) | (A!4 = empty_set)) <=> subset(A!4, unordered_pair(B!3, C!2)))) <=> ((~![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))) | (((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> subset(A!4, unordered_pair(B!3, C!2))))),
% 0.22/0.41 inference(transitivity,[status(thm)],[47, 45])).
% 0.22/0.41 tff(49,plain,
% 0.22/0.41 ((~![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))) | (((A!4 = unordered_pair(B!3, C!2)) | (A!4 = singleton(C!2)) | (A!4 = singleton(B!3)) | (A!4 = empty_set)) <=> subset(A!4, unordered_pair(B!3, C!2)))),
% 0.22/0.41 inference(quant_inst,[status(thm)],[])).
% 0.22/0.41 tff(50,plain,
% 0.22/0.41 ((~![A: $i, B: $i, C: $i] : (((A = unordered_pair(B, C)) | (A = singleton(C)) | (A = singleton(B)) | (A = empty_set)) <=> subset(A, unordered_pair(B, C)))) | (((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> subset(A!4, unordered_pair(B!3, C!2)))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.22/0.41 tff(51,plain,
% 0.22/0.41 (((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> subset(A!4, unordered_pair(B!3, C!2))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[50, 44])).
% 0.22/0.41 tff(52,plain,
% 0.22/0.41 ((~(((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> subset(A!4, unordered_pair(B!3, C!2)))) | (~((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2)))) | subset(A!4, unordered_pair(B!3, C!2))),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(53,plain,
% 0.22/0.41 ((~((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2)))) | subset(A!4, unordered_pair(B!3, C!2))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[52, 51])).
% 0.22/0.42 tff(54,plain,
% 0.22/0.42 ($false),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[53, 31, 19])).
% 0.22/0.42 tff(55,plain,(set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set), inference(lemma,lemma(discharge,[]))).
% 0.22/0.42 tff(56,plain,
% 0.22/0.42 ((((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> (set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)) | (~((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2)))) | (~(set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(57,plain,
% 0.22/0.42 ((~((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2)))) | (~(set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[56, 16])).
% 0.22/0.42 tff(58,plain,
% 0.22/0.42 (~((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2)))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[57, 55])).
% 0.22/0.42 tff(59,plain,
% 0.22/0.42 ((~((set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set) <=> subset(A!4, unordered_pair(B!3, C!2)))) | (~(set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)) | subset(A!4, unordered_pair(B!3, C!2))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(60,plain,
% 0.22/0.42 ((~(set_difference(A!4, unordered_pair(B!3, C!2)) = empty_set)) | subset(A!4, unordered_pair(B!3, C!2))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[59, 28])).
% 0.22/0.42 tff(61,plain,
% 0.22/0.42 (subset(A!4, unordered_pair(B!3, C!2))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[60, 55])).
% 0.22/0.42 tff(62,plain,
% 0.22/0.42 ((~(((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) <=> subset(A!4, unordered_pair(B!3, C!2)))) | ((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) | (~subset(A!4, unordered_pair(B!3, C!2)))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(63,plain,
% 0.22/0.42 (((A!4 = empty_set) | (A!4 = singleton(B!3)) | (A!4 = singleton(C!2)) | (A!4 = unordered_pair(B!3, C!2))) | (~subset(A!4, unordered_pair(B!3, C!2)))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[62, 51])).
% 0.22/0.42 tff(64,plain,
% 0.22/0.42 ($false),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[63, 61, 58])).
% 0.22/0.42 % SZS output end Proof
% 0.22/0.42 % E exiting
%------------------------------------------------------------------------------